Delay-independent stability from an undelayed linearization
= Delay-independent stability from an undelayed linearization
For $x'(t)=\alpha x(t-T)[1-x(t)]$, linearization about $x=1$ is $u'=-\alpha u$ because the delayed perturbation appears only in a quadratic term. The equilibrium is therefore locally asymptotically stable for every delay $T\geq0$.